Every conformal net has an associated unitary VOA

TL;DR

This paper proves that every conformal net has an associated unitary VOA and establishes a link between their representations, advancing the mathematical understanding of two-dimensional chiral conformal field theories.

Contribution

It demonstrates the existence of a unitary VOA for every conformal net and connects their representation theories, addressing a key part of the conjectured equivalence.

Findings

Every conformal net has an associated unitary VOA.

Representations with discrete spectrum and finite-dimensional eigenspaces yield VOA modules.

Advances the mathematical foundation of conformal field theory.

Abstract

Unitary vertex operator algebras (VOAs) and conformal nets are the two most prominent mathematical axiomatizations of two-dimensional unitary chiral conformal field theories. They are conjectured to be equivalent, but a rigorous comparison has proven challenging. We resolve one direction of the conjecture by showing that every conformal net has an associated unitary VOA. We also show that every representation of a conformal net in which the generator of rotation acts with discrete spectrum and finite-dimensional eigenspaces yields a unitary module of the corresponding VOA. A talk describing our results is available at: https://www.youtube.com/watch?v=f_LhNSeiiaE .